I wondered what the smallest prime of the form $2n^n+k$ is for some odd $k$. For $k<91$, there are small primes, but for $k=91$ , the smallest prime (if it exists) must be very large.

  • What is the smallest prime of the form $2n^n+91$ ?

It is clear that $\gcd(91,n)=1$ must hold. $2n^n+91$ is composite for every natural number $n$ below $1000$.

$$2\times 15^{15}+91 = 42846499\times 20440124659$$

shows that the smallest prime factor can be large.

  • $\begingroup$ No primes up to $n=1200$. Watch this space... $\endgroup$ – TonyK May 31 '15 at 11:43
  • $\begingroup$ Did you also check the range $0\le n \le 1000$ ? $\endgroup$ – Peter May 31 '15 at 11:46
  • $\begingroup$ Yes.${}{}{}{}{}{}$ $\endgroup$ – TonyK May 31 '15 at 11:47
  • $\begingroup$ No primes up to $n=1500$... $\endgroup$ – TonyK May 31 '15 at 11:53
  • $\begingroup$ I experience a deja-vu... There must be a glitch in the Matrix ! $\endgroup$ – Lucian May 31 '15 at 12:35

$$2 \times 1949^{1949} + 91$$

is probably prime! (Running a rigorous primality test on it would take almost a whole day $-$ see here, for instance $-$ so I'm not going to do that.) $2n^n+91$ is composite for all lower values of $n$.

  • $\begingroup$ How did you conclude that it is 'probably prime'? But it does look suspiciously prime... $\endgroup$ – Trogdor May 31 '15 at 12:45
  • $\begingroup$ @Trogdor there are plenty of test that show a number is composite, if they fail it is "probably prime." See en.wikipedia.org/wiki/Probable_prime $\endgroup$ – quid May 31 '15 at 12:59
  • $\begingroup$ I like "suspiciously prime". All we have to do is decide on a meaning for it. $\endgroup$ – TonyK May 31 '15 at 13:05
  • $\begingroup$ You're the one who made me recant my skeptical statements about proving a large number prime. Same person asking, too. They deleted my revised answer, which is a shame. math.stackexchange.com/questions/402357/… $\endgroup$ – Will Jagy May 31 '15 at 15:32
  • $\begingroup$ @WillJagy: Philistines. $\endgroup$ – TonyK May 31 '15 at 16:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.